Metric Regularity of Mappings and Generalized Normals to Set Images

Abstract

Abstract. The primary goal of this paper is to study some notions of normals to nonconvex sets in finite-dimensional and infinite-dimensional spaces and their images under single-valued and set.-valued mappings. The main motivation for our study comes from variational analysis and optimization. ·where the problems under consideration play a crucial role in many important aspects of generalized differential calculus and applications. Our major results provide precise equality formulas (sometimes just efiicient upper estimates) allowing us to compute generalized normals in various senses to direct and inverse images of nonconvex sets under single-valued and set-valued mappings between Banach spaces. The main tools of our analysis revolve around variational principles and the fundamental concept of metric regularity properly modified in this paper.

Second-Order Variational Analysis in Conic Programming with Applications to Optimality and Stability

“…Observe that Theorem 4.1 holds as formulated in arbitrary Banach spaces, where the notion of nondegeneracy is taken from [4,Definition 4.70] without assuming the finite-dimensionality of the spaces in question. Indeed, the only change in the proof given above is to replace the application of the finite-dimensional result from [40,Theorem 6.43] by its Banach space counterpart from [25,Theorem 4.2] ensuring the equality in (4.5) and hence in (4.1) under the surjectivity assumption imposed on ∇g(ȳ). In this way we can also derive the Banach space version of second-order chain rule from [35,Theorem 7] for the limiting constructions.…”

On the quantitative solution stability of parameterized set-valued inclusions

“…If Φ : X ⇒ Y is Lipschitz l.s.c. at (x, ȳ), then its inverse mapping Φ −1 : Y ⇒ X is hemiregular (alias, semiregular) at (ȳ, x), as understood in [5,13,15,21].…”

On the weak-star extensibility

“…The part (i) ⇒(ii) appeared in [1] (see also [2]) with basically the same proof as below. It is an integrated part in deriving generalized differential calculus rules involving direct and inverse images under the restrictive metric regular mappings (see [1,3,2]). Note that this result is similar to the so-called lifting property of sequences as in [9] (Section 2.4); see Theorem 2.4f therein.…”

“…Recently, this concept was also successfully employed to obtain more normal cone relations involving direct set images in [3].…”